LMA - Laboratoire de Mécanique et d’Acoustique

[Jeunes chercheurs] M. Volpe et C. Saade

Amphithéâtre François Canac, LMA

Le 7 juillet 2020 de 10h30 à 12h00

- Christelle Saade (doctorante équipe M&S)
- Marion Volpe (doctorante équipe Sons)

Christelle Saade

Space-time isogeometric methods for multi-field problems in mechanics

Abstract : Engineering and applied mathematics problems can be solved by many numerical methods. One of these methods is the widely adopted finite element method (FEM) [1].The finite elements method is generally applied in the discretization of space and often associated to a time stepping method such as the finite differences for the time resolution. Space-time discretization consists of considering time dependance as spatial dependance and a space-time variational principle is used. The very first publications on space-time methods began to show in the late 60s, one of them are by Oden [3] who employed the space-time method using Hamilton’s principle for dynamics. While the finite element method is widely used and proved to be effcient in lots of mathematical physics and engineering problems, it has also some limitations. In fact, in this method in general the solution is built with a low order continuity, typically C0-continuous interpolation on the boundary of the element. The isogeometric analysis(IGA) was recently introduced by Hughes and al. in 2005 [2] as a numerical method that offers real perspectives in the integration of geometric and computational models. It uses the same basis functions (B-splines and NURBS) for both describing the geometry of the computational domain and approximating the solution. It allows, on one hand, to exactly represent complex geometries, and on the other hand, to enrich the approximation basis compared to classical Lagrange polynomials (higher order of continuity between elements) which is highly important due to the fact that complex geometries can be made and represented in CAD design tools. The objective of this work is to take benefit of these properties to extend the solution in space, as classically associated to a finite differences method in time, to the space-time domain and to evaluate isogeometric analysis for space-time problems, i.e. to discretize both, the space and the time with isogeometric analysis. The space-time IGA is used for the resolution of elastodynamics and multiphysics problems. Linear and nonlinear equations are treated, small and finite strains are considered in our study. We solve the problems in 1D and in 2D, and optimal convergence rates are obtained, showing that space-time methods offer interesting perspectives for the resolution of various numerical problems in mechanics. We expect from these methods more stable and robust schemes compared to standard schemes such as the HHT or Newmark type methods, and larger time steps employed in discretization.

[1] Zienkiewicz, O.C. and Taylor, R.L. The finite element method. McGraw Hill, Vol. I., (1989), Vol. II., (1991).
[2] Hughes, T.J.R., Cotrell, J.A. and Bazilevs, Y. Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, Vol. 194, 39-41:4135-4195(2005).
[3] Oden, J. Tinsley A general theory of finite elements. II. Applications. International Journal for Numerical Methods in Engineering, (1969).